We give a combinatorial interpretation of the determinant of a matrix as a
generating function over Brauer diagrams in two different but related ways. The sign of a
permutation associated to its number of inversions in the Leibniz formula for the
determinant is replaced by the number of crossings in the Brauer diagram. This
interpretation naturally explains why the determinant of an even antisymmetric matrix is
the square of a Pfaffian.

We define a de Bruijn process with parameters n and L as a certain continuous-time
Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as
vertices. We determine explicitly its steady state distribution and its characteristic
polynomial, which turns out to decompose into linear factors. In addition, we examine the
stationary state of two specializations in detail. In the first one, the de
Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de
Bruin process, the distribution has constant density but nontrivial correlation functions.
The two point correlation function is determined using generating function techniques.

The refined enumeration of alternating sign matrices (ASMs) of given order having
prescribed behavior near one or more of their boundary edges has been the subject of
extensive study, starting with the Refined Alternating Sign Matrix Conjecture of
Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and
triply-refined enumeration by several authors. In this paper we extend the previously known
results on this problem by deriving explicit enumeration formulas for the "top-left-bottom"
(triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter
case solves the problem of computing the full boundary correlation function for ASMs. The
enumeration formulas are proved by deriving new representations, which are of independent
interest, for the partition function of the square ice model with domain wall boundary
conditions at the "combinatorial point" 2{\pi}/3.

In this paper, we propose a new Markov chain which generalizes
random-to-random shuffling on permutations to random-to-random shuffling on
linear extensions of a finite poset of size $n$. We conjecture that the second
largest eigenvalue of the transition matrix is bounded above by
$(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov
chain provides a way to sample the linear extensions of the poset with a
relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log
n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the
usual random-to-random shuffling.

We demonstrate a natural bijection between a subclass of alternating sign matrices
(ASMs) defined by a condition on the corresponding monotone triangle which we call the
gapless condition and a subclass of totally symmetric self-complementary plane partitions
defined by a similar condition on the corresponding fundamental domains or Magog triangles.
We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding
permutations. This leads us to generalize pattern avoidance on permutations to a family of
words associated to ASMs, which we call Gog words. We translate the gapless condition on
monotone trangles into a pattern avoidance-like condition on Gog words associated. We
estimate the number of gapless monotone triangles using a bijection with p-branchings.

We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.

We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset. This gives rise to a strongly
connected graph on L. In earlier work (arXiv:1205.7074), we studied
promotion-based Markov chains on these linear extensions which generalizes
results on the Tsetlin library. We used the theory of R-trivial monoids in an
essential way to obtain explicitly the eigenvalues of the transition matrix in
general when the poset is a rooted forest. We first survey these results and
then present explicit bounds on the mixing time and conjecture eigenvalue
formulas for more general posets. We also present a generalization of promotion
to arbitrary subsets of the symmetric group.

Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$. We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.

We define two general classes of nonabelian sandpile models on directed trees
(or arborescences) as models of nonequilibrium statistical phenomena. These
models have the property that sand grains can enter only through specified
reservoirs, unlike the well-known abelian sandpile model.
In the Trickle-down sandpile model, sand grains are allowed to move one at a
time. For this model, we show that the stationary distribution is of product
form. In the Landslide sandpile model, all the grains at a vertex topple at
once, and here we prove formulas for all eigenvalues, their multiplicities, and
the rate of convergence to stationarity. The proofs use wreath products and the
representation theory of monoids.

We develop a general theory of Markov chains realizable as random walks on
$\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues
of the transition matrix, for multiplicities of the eigenvalues via M\"obius inversion
along a lattice, a condition for diagonalizability of the transition matrix and some
techniques for bounding the mixing time. In addition, we discuss several examples, such as
Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples
previously studied by the authors, such as nonabelian sandpile models and the promotion
Markov chain on posets. Many of these examples can be viewed as random walks on quotients
of free tree monoids, a new class of monoids whose combinatorics we develop.